

A139005


Primes p such that there exist primes p'<p"<p"'<p""<p such that the concatenation of any two among the {p,...,p""} is prime.


0



8389, 18433, 25253, 31231, 33647, 40289, 40357, 47237, 47303, 48731, 51721, 55621, 57331, 58763, 61129, 62303, 63601, 64189, 65657, 65677, 65983, 67723, 68491, 70099, 70571, 71341, 71741, 72739, 75653, 77153, 77641, 78509, 78511, 81401
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OFFSET

1,1


COMMENTS

Obviously, 2 and 5 cannot be part of such a 5tuple.
Apart from 3, all the elements of the quintuple must have the same
residue mod 3, thus they are all = 1 or = 1 mod 6.
Two curious facts can be observed: for p=51721 and p=63601, there are
two possibilities for (p',p",p"',p""), which differ in both cases only
in p"": [p=51721, p""=44371 or 44683, p"'=22531, p"=12703, p'=8101],
resp. [63601, 26893 or 61417, 25939, 61, 7]. Secondly, the two
quintuples for p=51721 are followed by a quintuple whose sum is one of
smallest possible, only roughly half that of the two preceding
solutions.


LINKS



EXAMPLE

The second such 5tuple is [18433, 12409, 2341, 1237, 7].
The two quintuplets [51721, 44371, 22531, 12703, 8101]
/*sum=139427*/ and [51721, 44683, 22531, 12703, 8101] /*sum=139739*/
are followed by [55621, 18493, 991, 883, 733] /*sum=76721*/.
The next "degenerate" case is [63601, 26893, 25939, 61, 7]
/*sum=116501*/ and [63601, 61417, 25939, 61, 7] /*sum=151025*/.
The third "degenerate" case is [71341, 63277, 54583, 7741,
241] /*sum=197183*/ and [71341, 63277, 54583, 36187, 241]
/*sum=225629*/.


PROG

(PARI) /* available from the author upon solving "Problem 60" on ProjectEuler.net */


CROSSREFS



KEYWORD

nonn,base


AUTHOR



STATUS

approved



